Combined and Comparative Analysis of Power Spectra

In solar physics, especially in exploratory stages of research, it is often necessary to compare the power spectra of two or more time series. One may, for instance, wish to estimate what the power spectrum of the combined data sets might have been, or one may wish to estimate the significance of a particular peak that shows up in two or more power spectra. One may also on occasion need to search for a complex of peaks in a single power spectrum, such as a fundamental and one or more harmonics, or a fundamental plus sidebands, etc. Visual inspection can be revealing, but it can also be misleading. This leads one to look for one or more ways of forming statistics, which readily lend themselves to significance estimation, from two or more power spectra. We derive formulas for statistics formed from the sum, the minimum, and the product of two or more power spectra. A distinguishing feature of our formulae is that, if each power spectrum has an exponential distribution, each statistic also has an exponential distribution. The statistic formed from the minimum power of two or more power spectra is well known and has an exponential distribution. The sum of two or more powers also has a well-known distribution that is not exponential, but a simple operation does lead to an exponential distribution. Concerning the product of two or more power spectra, we find an analytical expression for the case n = 2, and a procedure for computing the statistic for n > 2. We also show that some quite simple expressions give surprisingly good approximations.